Final answer:
The question involves the standard normal distribution and the application of a z-table, used to find the area under the curve corresponding to specific z-scores, which in turn determines probabilities and percentiles for a given distribution.
Step-by-step explanation:
Understanding the Standard Normal Distribution
The question discusses the concept of the standard normal distribution, a fundamental concept in statistics used to describe data that is symmetrically distributed around a mean of zero and a standard deviation of one. When the question refers to shading the region under the curve to the left of x = 0.00, it is asking to visualize the portion of the curve that represents all values below the mean. Since the total area under the curve is 1 (or 100%), the area to the left of x = 0.00 represents half of the distribution, or 0.5000.
The use of a z-table is also mentioned, which is a tool for finding the area to the left of a given z-score. This area reflects the probability that a random variable X is less than that z-score. For a z-score of 0.00, the area to the left is typically 0.5000, since it is the midpoint of the normal distribution. We use the z-table to find areas under the curve for other z-scores as well, either directly or by subtracting from 1 when we need the area to the right of the z-score.
Additionally, understanding these areas is crucial when calculating probabilities for certain values of X. For example, if the question asks for the probability that X is less than a certain value, we would convert that value to a z-score and then use the z-table to find the corresponding area. Similarly, to find percentile values such as the 80th or 60th percentile, we identify the z-score such that the area to the left of it is 0.80 or 0.60, respectively, and then convert this z-score back to the original measurement using the mean and standard deviation provided.